Neurocomputing 70 (2006) 206–218 Balanced parameterization of multichannel blind deconvolutive systems: A continuous time realization Liangsuo Ma a,Ã , Ah Chung Tsoi b a Department of Radiology, University of Iowa, Iowa City, IA 52246, USA b Monash University, Clayton Campus, Wellington Road, Clayton, Vic. 3168, Australia Received 3 December 2004; received in revised form 31 December 2005; accepted 24 March 2006 Communicated by S. Hochreiter Available online 7 July 2006 Abstract A key problem in multichannel blind deconvolution (MDB) is how to properly model the mixer so that the demixer can be properly modelled correspondingly. This problem naturally triggers one question, viz. how to determine some key model orders, e.g. the number of states, if the problem is analyzed using a state space model. In this paper, to answer this question, we will apply a balanced parameterization approach to a MBD problem with the mixer being modelled as a continuous time linear time invariant (LTI) system. Besides allowing the determination of the number of states required in the demixer, compared with the controller canonical form representation or the observer canonical form representation of the LTI continuous time system, our approach has also the advantages of numerical robustness. The proposed method is validated through practical examples using speech signals and electroencephalographic (EEG) data. r 2006 Elsevier B.V. All rights reserved. Keywords: Blind source separation; Multichannel blind deconvolution; Independent component analysis; State space; Balanced parameterization 1. Introduction The multichannel blind deconvolution (MBD) problem can be formulated as follows: given a set of observation data: the sensor outputs of an unknown multi-input and multi-output dynamical system, the objective of MBD is to recover the latent input signals (sources) based on the output observations. Generally, the inputs are generated by a number of unknown statistically independent signals. In other words, the observation data are the convolutive mixture of the latent sources and the unknown dynamical system. In the context of MBD, the unknown dynamical system is usually called the mixer. Further, the mixer is assumed to be a linear time invariant (LTI) dynamical system in this paper. 1 The MBD problem is usually solved by devising an inverse system of the mixer, which is generally referred as a demixer. A key problem in MBD is how to properly model the mixer so that the demixer can be modelled correspond- ingly. This problem naturally triggers one question, viz. how to determine some key model orders of the system models. First, we need to decide if the problem is in the continuous time domain or the discrete time domain. Dependent on the situation, one may assume that the mixer to be either a continuous time dynamical system, or a discrete time dynamical system. In this paper, we will assume that the mixer is modelled by a continuous time dynamical system. Accordingly the demixer is also modelled by a continuous time dynamical system. 2 ARTICLE IN PRESS www.elsevier.com/locate/neucom 0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.neucom.2006.03.011 Ã Corresponding author. E-mail addresses: liangsuo-ma@uiowa.edu (L. Ma), ahchung.tsoi@adm.monash.edu.au (A.C. Tsoi). 1 In general, the mixer can be a nonlinear time varying system. In this paper, we will only consider a LTI mixer. This may be considered as the (footnote continued) simplest possible dynamical system representing a dynamic mixer situation. 2 Note that this is not necessarily the case all the time. It may be possible to have a discrete time demixer to a continuous time mixer, if the sampling frequency is sufficiently high. On the other hand, it is also possible to have a continuous demixer to a discrete time mixer, provided that the sampling